barcode reader integration with asp.net A- = aiY in Software

Creator Code 128 Code Set A in Software A- = aiY

(28A. 1)

where IAl is the determinant of A and adj A is the adjoint of A. These two terms will now be described . The determinant of a matrix IAl is a scalar which is computed from the elements of the matrix as follows:

(28A.2)

where Aij , the cofactor of the element a ij, is computed as Aij = (-l) +jMij The determinant Mij is the minor of the element aij and is defined as follows. If the row and column containing the element a ij are deleted from a square matrix A, the determinant of the resulting matrix, which is an (n - 1) X (n - 1) matrix, is the minor Mij . An alternate expression for the calculation of a determinant which uses the elements of a specific column and its cofactors is as follows: IAl = TaijAij

(28A.3)

We now define the adjoint of a matrix. Let the matrix B be an IZ X n matrix whose elements bij are the cofactors Aji of A, i.e., the transpose of the cofactor matrix. B is the adjoint of A; thus B = adj A = transpose of cofactor matrix or 41 A21 . . . An1

adj A = *

. . .

. . .

. . .

The derivations of relationships presented here, as well as other properties of matrices, can be found in textbooks on linear algebra (see Anton, 1984).

For this problem, we use Eq. (28A.2) with i = 1 (i.e., use row 1)

JAI = 2[; 3 -3[i A] +q; ;]

2UNO)

IAl = 2(-1) - 3(-2) + 5(1) = 9

2 3 1 4

The adjoint of the matrix, which is the transpose of the cofactor matrix, is adjA = [-: therefore A-l = i [ -: -z] = [-t 3. Obtain the inverse of the matrix 2 3 A = One can show that IA/ = 18 The cofactor matrix is * -5 7 The adjoint matrix is r 1 7 .,-: A- = ; 7 I -5 -5 1 -5 7 1 7 7-5 1 7 -5 1 1 7 1 -5 -57 1 1 2 3 1 13 2 -i]

28.1. In the liquid level process shown in Fig. p28.1, the three tanks are interacting. The process may be described by: k=Ax+Bu

---3

A=2-3 0 1

determine values of R3, Rq, and Rs. If one of these values of R is negative, what is your interpretation (b) Determine B. 28.2. For the system shown in Fig. P28.2, find A and b in

FIGURE P28-2

In the previous chapter, we have seen that a linear dynamic system can be expressed in terms of the following equations i = Ax+Bu y = cx where x = column vector of n state variables (X 1~2, . . . ,x,) u = column vector of m inputs or forcing terms (U t ,U 2, . . . ,u m) y = column vector of p outputs (yi,y2, , . . ,y,) A = n X n matrix of coefficients B = it X m matrix of coefficients C = p X n matrix of coefficients One of the objectives of this chapter is to show how one solves Eqs. (29.1) and (29.2) in a systematic manner. Before discussing the solution of the matrix differential equation of Eq. (29. l), consider the scalar differential equation (29.3) dxldt = Ax + Bu In this equation all of the terms are scalars. The solution to Eq. (29.3) can be written as the sum of the complementary function and the particular integral as follows:

x(t)

(29.1) (29.2)